Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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10 Most Recent Blog Comments:

Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

August 25, 2019 at 12:21am

#964690

Professional Crastination

This one's been in my blog fodder queue for a long time, and I'm finally getting around to it.

https://www.fastcompany.com/90357248/procrastination-is-an-emotional-problem

Procrastination is an emotional problem
If you stop treating procrastination like a time-management issue, it becomes easier to manage.

I haven't made it a secret that I have a problem with procrastination. It's not as bad now as it used to be - perhaps because a daily exercise routine has taken the edge off my depression, at least for now. It'll come back. But what made me finally start exercising in the first place? I don't know.

This emotional avoidance technique that our brain–often subconsciously–employs is similar to that which underlies many types of anxiety. People with anxiety often do everything they can to avoid the perceived external threat and, in turn, shut off access to both good and bad feelings, often leading to depression. By procrastinating, we’re avoiding a task with the assumption that the task won’t feel good, and that means we’re missing out on any feelings of, for example, accomplishment or success. This connection between procrastination and depression has been around at least since the ’90s, and the experimental evidence has poured in ever since.

Yeah, I'm not sure that makes any sense. Sure, we think the task, whatever it is, won't feel good - but we know that procrastinating can only make it worse.

“Giving in to feel good” is the term given to this phenomenon in one paper cited by many procrastination researchers. And it means seeking short-term good feelings at the cost of long-term satisfaction–something we’re known to do as early as toddlerhood.

I think one of the greatest decisions I ever made was to give in to the short-term at the expense of the long-term. It works for me now.

Let’s start with the relationship between self-compassion and procrastination because it’s both counterintuitive and revealing. What’s the first thing you do when you catch yourself indulging in a particularly egregious spell of procrastination? Do you tell yourself, “What’s wrong with you? Pull yourself together and get your work done!” That lack of self-compassion might be exactly what’s causing your procrastination in the first place, according to the research.

This, on the other hand, rings true for me. When I'd procrastinate, I'd mentally beat myself up. "Just do it! Just fucking work! What the hell, Waltz?" Or I'd imagine someone else saying that to me, which doesn't help either.

Another study found that procrastination is “associated significantly with negative automatic thoughts in general as well as automatic thoughts reflecting the need to be perfect.” In both studies, this highly self-critical mind-set created and perpetuated the problem of procrastination.

I had a problem with that, too. "If it's worth doing, it's worth doing well" has a corollary: "If it can't be done perfectly, why bother?" That took me a while to shake.

The first, which comes from Buddhist psychology, is the idea of the “monkey mind” that we all share. “The monkey mind never stops and you can’t make it stop,” Pychyl says. “Instead, you’ve got to give the monkey something to do.”

The monkey has plenty to do. Video games, e.g. Wrong. Next.

The second idea, which comes from more traditional psychology, is that our emotions can’t be pushed aside or ignored. So when we have a strong aversion to getting our work done, we can’t ignore this feeling.

Right, like I could ignore it. It's right there in the forefront of my consciousness. Wrong. Next.

The third part comes from David Allen, the founder of the Getting Things Done™ method, which is the idea that we don’t do projects when we work; we do actions. In other words, the mountain of work that we picture ourselves wading through is really just a set of smaller, discrete actions that have to be taken one at a time. We put our pants on one leg at a time and write our articles one word at a time.

This has some ring of truth to it. Any project can be broken down into smaller and smaller sub-actions until you reach the equivalent of the subatomic level. Each indivisible sub-action, then, which I'll call an "acton" to keep with subatomic nomenclature like proton or electron, is easy. It's only putting them together that's hard. The downside to this? A novel, for example, has millions or billions of actons: each strike of a key, each stray thought, every website visited for research, every check of a thesaurus, each mental review of upcoming dialogue. And millions as a number is hard to contemplate, so we usually break it down to words, of which there are only thousands. And then, for something like NaNoWriMo, you might set a goal of 1667 words a day, which feels manageable. Usually.

“Willpower” is a slippery concept. Some researchers believe it doesn’t really exist. Some believe it exists but in finite supply. Others take a middle path.

I'm in the "doesn't really exist" camp. There's an article about that I'll share at some point. In short, though, while it may sometimes seem that mind and body are separate, or that there are competing ideas in "mind," we're really just one unified physical entity. So there's nothing separate from us to provide the "willpower."

Instead, he recommends cultivating another mental skill: mindfulness.

The skills developed in mindfulness meditation, such as concentration, non-judgment, and equanimity, align perfectly with the research showing the vital role of emotional regulation in reducing procrastination and improving productivity.

Sure, if you have the patience for that "mindfulness" crap. Maybe you do and it's not crap to you. But I'd rather do my taxes or clean my bathroom than set aside time for meditation.

Consequently, maybe it would work for me after all: by doing the dreaded task, whatever it is, I can use it to procrastinate mindfulness meditation. Win/win.

One of these days, I'm going to host a procrastinators convention.

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